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\begin{document}

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\section*{ZFA}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{zermelofraenkel_set_theory_with_atoms}{}\section*{{Zermelo--Fraenkel set theory with atoms}}\label{zermelofraenkel_set_theory_with_atoms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{empty_atoms}{Empty atoms}\dotfill \pageref*{empty_atoms} \linebreak
\noindent\hyperlink{reflexivequine_atoms}{Reflexive/Quine atoms}\dotfill \pageref*{reflexivequine_atoms} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{fraenkelmostowski_models}{Fraenkel--Mostowski models}\dotfill \pageref*{fraenkelmostowski_models} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{ZFA} is a variant of the [[material set theory]] [[ZF]] which allows for objects, called \emph{atoms} or \emph{[[urelements]]}, which may be members of [[sets]], but are not made up of other elements. ZFA featured in early independence proofs, notably [[Fraenkel-Mostowski permutation models]], for example showing AC is independent of the rest of the axioms of ZFA.

[[Zermelo]]`s original 1908 axiomatisation of set theory included atoms, but they were soon discarded as a foundational approach as they could be modeled inside of atomless set theory.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are two possible approaches to formulating ZFA. In both cases, we can further require that the [[axiom of choice]] is satisfied, and obtain ZFCA.

\hypertarget{empty_atoms}{}\subsubsection*{{Empty atoms}}\label{empty_atoms}

In this approach, atoms are empty. We start by adding an additional unary predicate $A$, where we interpret $A(x)$ as saying ``$x$ is an atom''. We write $(\forall set x)$ to mean $\forall x, \neg A(x) \Rightarrow$, and similarly write $(\forall atom x)$ to mean $\forall x, A(x) \Rightarrow$.

Then the [[axiom of extensionality]] says

\begin{displaymath}
(\forall set x) (\forall set y) (\forall z, z \in x \Leftrightarrow z \in y)  \Rightarrow x = y,
\end{displaymath}
and the [[axiom of empty set]] says

\begin{displaymath}
(\exists set x) (\forall y) (y \notin x).
\end{displaymath}
We also add the axiom that says atoms are empty:

\begin{displaymath}
(\forall atom a) (\forall x) x \notin a.
\end{displaymath}
Sometimes it is also convenient to assume that we have a set of atoms:

\begin{displaymath}
(\exists X) (\forall x) (A(x) \Leftrightarrow x \in X),
\end{displaymath}
but in some cases, we might also like to consider models with a proper class of atoms.

\hypertarget{reflexivequine_atoms}{}\subsubsection*{{Reflexive/Quine atoms}}\label{reflexivequine_atoms}

We can give up on the [[axiom of foundation]], and introduce the urelements as [[reflexive sets]], ie. sets $x$ such that $x = \{x\}$. In place of the axiom of foundation, we can have an axiom of \emph{weak} foundation, where we require the existence of a set A such that every element of A is reflexive, and the [[cumulative hierarchy]] built up from $A$ is the whole universe. In other words, if we define

\begin{itemize}%
\item $R(0) = A$,


\item $R(\alpha + 1) = P(R(\alpha))$ for any ordinal $\alpha$,


\item $R(\lambda) = \bigcup_{\gamma \lt \lambda} R(\gamma)$ for $\lambda$ a limit ordinal,



\end{itemize}
then $V = \bigcup_\alpha R(\alpha)$.

\hypertarget{models}{}\subsection*{{Models}}\label{models}

\hypertarget{fraenkelmostowski_models}{}\subsubsection*{{Fraenkel--Mostowski models}}\label{fraenkelmostowski_models}

By allowing atoms in our models, we lend ourselves to the method of [[Fraenkel-Mostowski models]], where we can obtain models in which the [[axiom of choice]] fails by imposing some symmetry on the atoms (so that we cannot uniformly pick an atom out of many). Such models are closely related to [[categories of G-sets]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[ZFC]]
\item [[Fraenkel-Mostowski models]]
\item [[permutation model]]
\item [[urelements]]

\end{itemize}
[[!redirects ZFA]] [[!redirects ZFU]] [[!redirects ZFCA]] [[!redirects ZFAC]]



\end{document}